LIST OF FIGURES

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Regional Flood Frequency Analysis for Newfoundland and Labrador Using the L-Moments Index-Flood Method

By

Yang Lu

A thesis submitted to the School of Graduate Studies in partial fulfillment of the requirements for the degree of

Master of Engineering

Faculty of Engineering and Applied Science Memorial University of Newfoundland

May 2016

St. John’s Newfoundland Canada

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ABSTRACT

The L-moments based index-flood procedure had been successfully applied for Regional Flood Frequency Analysis (RFFA) for the Island of Newfoundland in 2002 using data up to 1998. This thesis, however, considered both Labrador and the Island of Newfoundland using the L-Moments index-flood method with flood data up to 2013. For Labrador, the homogeneity test showed that Labrador can be treated as a single homogeneous region and the generalized extreme value (GEV) was found to be more robust than any other frequency distributions. The drainage area (DA) is the only significant variable for estimating the index-flood at ungauged sites in Labrador.

In previous studies, the Island of Newfoundland has been considered as four homogeneous regions (A, B, C and D) as well as two Water Survey of Canada’s Y and Z sub-regions. Homogeneous regions based on Y and Z was found to provide more accurate quantile estimates than those based on four homogeneous regions.

Goodness-of-fit test results showed that the generalized extreme value (GEV) distribution is most suitable for the sub-regions; however, the three-parameter lognormal (LN3) gave a better performance in terms of robustness. The best fitting regional frequency distribution from 2002 has now been updated with the latest flood data, but quantile estimates with the new data were not very different from the previous study.

Overall, in terms of quantile estimation, in both Labrador and the Island of Newfoundland, the index-flood procedure based on L-moments is highly recommended as it provided consistent and more accurate results than other techniques such as the regression on quantile technique that is currently used by the government.

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ACKNOWLEDGEMENTS

I am grateful thankful to my parents for their endless love, understanding and support they gave me throughout my studies and my life.

I would also like to thank to my supervisor, Dr. Leonard M. Lye, who provided me with financial help and academic guidance in the process of conducting this research and writing this thesis.

Thanks also go the School of Graduate Studies of Memorial University and to the Institute for Biodiversity and Environmental Sustainability (IBES) for providing partial funding for the study.

I am also grateful to Ms. Jinghua Nie for her sincere help for my life and study.

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LIST OF FIGURES

Figure 3.1 L-moment ratio diagram (after Hosking and Wallis, 1997) ... 30

Figure 4.1 Location of studied sites in Labrador (Cited and modified based on the report conducted by AMEC, 2014) ... 50

Figure 4.2 Boxplot of site 03OC003 ... 51

Figure 4.3 L-moment ratios in Labrador ... 53

Figure 4.4 L-moment ratio diagram in Labrador ... 57

Figure 4.5 Regional GEV growth curve for Labrador with 90% confidence intervals ... 63

Figure 4.6 Regional quantile function fit observed data with 90% confidence limits in Labrador ... 64

Figure 4.7 Regression relationship of Drainage Area (DA) vs. Peak Flow (Q)... 69

Figure 4.8 Regional growth factor has a good agreement with the observed data .... 71

Figure 4.9 Comparison of quantile estimates for Q50 and Q100 between at-site and regional analysis based on the L-moments based index-flood procedure ... 73

Figure 4.10 Comparison of quantile estimates for Q50 and Q100 between at-site and regional analysis based on the regression on quantiles method obtained from AMEC (2014) ... 74

Figure 4.11 Comparison of quantile estimates between L-moments based index-flood procedure and quantile regression models (AMEC, 2014) in Labrador ... 77

Figure 5.1 Locations of studied sites in Newfoundland (Cited and modified based on the research conducted by Zadeh , 2012) ... 81

Figure 5.2 Boundary of sub regions Y and Z in Newfoundland (Cited and modified based on the research conducted by Zadeh , 2012) ... 82

Figure 5.3 Boundaries of sub regions A, B, C and D in Newfoundland (Cited and modified based on the research conducted by Zadeh , 2012) ... 83

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Figure 5.4 L-moment ratios plots in Newfoundland ... 91 Figure 5.5 Boxplot logged data of site 02ZM009 ... 97 Figure 5.6 Boxplot of site 02ZM010... 97 Figure 5.7 L-moment ratio diagram and regional L-moment ratios in Newfoundland ... 103 Figure 5.8 L-moment ratio diagram for sub region Y ... 135 Figure 5.9 L-moment ratio diagram for sub region Z ... 135 Figure 5.10 Regional frequency model has a good agreement with observed value149 Figure 5.11 Comparison of quantile estimates for Q50 and Q100 between at-site and regional analysis in four sub regions in Newfoundland ... 153 Figure 5.12 Comparison of quantile estimates between the index-flood procedure and regression models for each tested site in Newfoundland ... 156 Figure 5.13 Comparison of quantile estimates for Q50 and Q100 between at-site and regional analysis in Region Y ... 157 Figure 5.14 Comparison of quantile estimates for Q50 and Q100 between at-site and regional analysis in Region Z ... 157 Figure 5.15 Comparison of quantile estimates between the index-flood procedure and regression models for each tested site in Newfoundland ... 161

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LIST OF TABLES

Table 3.1 Critical values of discordancy measure with N sites (Hosking and Wallis,

1997) ... 34

Table 3.2 Polynomial approximations of τ4 as a function of τ₃ (Hosking and Wallis, 1997) ... 37

Table 4.1 Regression equations and goodness-of-fit developed by AMEC (2014) for Labrador ... 47

Table 4.2 Summary statistics, L-moment ratios and discordancy measure (Di) for 10 sites in Labrador ... 52

Table 4.3 Results of discordancy measure after removing site 03OC003 ... 54

Table 4.4 Weighted L-moment ratios, kappa parameters, μv, σv and H value of Labrador ... 55

Table 4.5 Weighted L-moment ratios, kappa parameters, μv, σv and H value in the absence of site 03OC003 ... 56

Table 4.6 Results of goodness-of-fit measure of five candidate distributions ... 58

Table 4.7 Results of robustness test in Labrador ... 60

Table 4.8 Regional GEV parameters and GEV quantile function in Labrador ... 62

Table 4.9 Return period growth factor with 90% confidence intervals in Labrador .. 64

Table 4.10 Results of comparison between at-site and regional quantile estimates in Labrador ... 65

Table 4.11 Results of comparison between at-site and regional quantile estimations in Labrador ... 67

Table 4.12 Results of comparison of quantile estimation based on the median and mean ... 68

Table 4.13 Basic information of the stations for verification in Labrador ... 70 Table 5.1 Basic information and L-moment ratios for sub region A in Newfoundland

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... 84 Table 5.2 Basic information and L-moment ratios for sub region B in Newfoundland ... 85 Table 5.3 Basic information and L-moment ratios for sub region C in Newfoundland ... 86 Table 5.4 Basic information and L-moment ratios for sub region D in Newfoundland ... 87 Table 5.5 Results of discordancy measure (Di) of studied sites in sub region A in Newfoundland ... 93 Table 5.6 Results of discordancy measure (Di) of studied sites in sub region B in Newfoundland ... 94 Table 5.7 Results of discordancy measure (D_i) of studied sites in sub region C in Newfoundland ... 95 Table 5.8 Results of discordancy measure (D_i) of studied sites in sub region D in Newfoundland ... 96 Table 5.9 Results of discordancy measure (Di) of 15 selected sites excluding 02ZM009 in sub region A ... 98 Table 5.10 Results of discordancy measure (Di) of 15 selected sites excluding 02ZM010 in sub region A ... 99 Table 5.11 Kappa parameters and results of the heterogeneity measure in Newfoundland ... 101 Table 5.12 Results of goodness-of-fit measure for sub regions in Newfoundland .. 105 Table 5.13 Results of robustness test for four sub regions in Newfoundland ... 107 Table 5.14 Regional parameters of the LN3 distribution and regional quantile functions for sub regions in Newfoundland ... 110 Table 5.15 Comparison of at-site and regional frequency estimation for sub region A ... 111 Table 5.16 Comparison of at-site and regional frequency estimation for sub region B

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... 112

Table 5.17 Comparison of at-site and regional frequency estimation for sub region C ... 113

Table 5.18 Comparison of at-site and regional frequency estimation for sub region D ... 114

Table 5.19 Regression equations and goodness-of-fit developed by AMEC (2014) in Newfoundland ... 115

Table 5.20 Comparison of at-site and regional quantile flows for sub region A ... 117

Table 5.21 Comparison of at-site and regional quantile flows for sub region B ... 118

Table 5.22 Comparison of at-site and regional quantile flows for sub region C ... 119

Table 5.23 Comparison of at-site and regional quantile flows for sub region D ... 120

Table 5.24 Nonlinear regression equations and R² for sub regions in Newfoundland ... 121

Table 5.25 Summary of statistics and discordancy measure of sub region Y ... 123

Table 5.26 Summary of statistics and discordancy measure of sub region Z ... 125

Table 5.27 Results of discordancy measure in sub region Z excluding site 02ZM009 ... 127

Table 5.28 Results of discordancy measure in sub region Z excluding site 02ZM010 ... 129

Table 5.29 Results of discordancy measure in sub region Z excluding sites 02ZM010 and 02ZM009 ... 131

Table 5.30 Results of heterogeneity measure for sub regions Y and Z ... 133

Table 5.31 Results of goodness-of-fit test for candidate distributions in sub regions Y and Z... 136

Table 5.32 Results of robustness test for sub regions Y and Z... 138

Table 5.33 Regional parameters of LN3 distribution and LN3 quantile functions in sub regions Y and Z ... 139 Table 5.34 Results of comparison between at-site and regional analysis in region Y

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... 140 Table 5.35 Results of comparison between at-site and regional analysis in region Z ... 141 Table 5.36 Comparison of at-site and regional frequency estimates between current research and Pokhrel’s research (2002) in sub region Y ... 142 Table 5.37 Comparison of at-site and regional frequency estimates between current research and Pokhrel’s research (2002) in sub region Z ... 143 Table 5.38 Comparison of regional frequency estimates for sub regions Y and Z .. 145 Table 5.39 Comparison of regional frequency estimates of studied sites in Newfoundland ... 147 Table 5.40 Basic information of verification stations in sub region Y in Newfoundland ... 149 Table 5.41 Flood information of gauged sites for verification ... 150 Table 5.42 Nonlinear regression equations and R²for sub regions Y and Z in Newfoundland ... 162

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LIST OF SYMBOLS

Q̅ Mean annual peak flow τ L-CV

µ Mean

σ Standard deviation

ξ Location parameter of the distribution α Scale parameter of the distribution ε Error

τ3 L-skewness (population)

τ4 Standard deviation of sample regional L-kurtosis

τ₄^DIST Distribution’s L-kurtosis

βr Population probability weighted moment λr Population L-moments

Ak Coefficients of Polynomial Approximations An Site Characteristics

B Bias

B4 Bias of sample regional L-kurtosis Di Discordancy measure

E Expected value

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F Non-exceedance probability F(x) Cumulative distribution function

H Heterogeneity measure

h 4^th parameter of the kappa distribution k Shape parameter of the distribution In Natural logarithm

lr Sample L-moments

Nsim Number of simulated regions Q Flow rate

q Quantile function t Sample L-CV

t3 Sample L-skewness t3R

Regional average sample L-skewness t4 Sample L-kurtosis

t₄R

Regional average sample L-kurtosis tR

Regional average sample L-CV µv Mean of simulated sites

V Weighted standard deviation of at-site sample L-CVs ZDIST

Goodness-of-fit measure of the candidate distribution

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LIST OF ACRNYMS

AARB Absolute Relative Bias ACLS Lakes and Swamps ARB Average Relative Bias

AMEC AMEC Environment & Infrastructure DA Drainage Area

DRD Drainage Density

GEV Generalized Extreme Value GLO Generalized Logistic GPA Generalized Pareto IFM Index-flood Method IH Institute of Hydrology LAF Lake Attenuation Factor L-CV Coefficient of L-variation LN3 3-parameter Log Normal LSF Lakes and Swamps Factor

NE Northeast

NERC Natural Environment Research Council

NW Northwest

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PE3 Pearson Type III

PWMs Probability Weighted Moments RFFA Regional Flood Frequency Analysis RMSE Root Mean Square Error

SE Southeast

SEE Standard Error of the Estimate SMR Regression Correlation Coefficient

SW Southwest

USGS United States Geological Survey WSC Water Survey of Canada

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CHAPTER 1

INTRODUCTION

1.1 General

Accurate estimations of flood quantiles play a significant role in minimizing flood damage, specifically related to casualties, compensation related expenses and environmental damage, which are all caused by flooding. Furthermore, accurate estimations of flood frequencies can provide valuable information for designing and planning hydraulic structures and other flood protection schemes.

Flood frequency analysis was traditionally based on fitting a frequency distribution or probability model to the observed flood data at a single site. However, insufficient data often create a challenge for hydrologists to provide an accurate flood quantile. A preferable approach is to use regional flood frequency analysis (RFFA) to deal with this problem. RFFA uses data at neighboring sites in a defined homogeneous region to develop a model. Flood quantiles at any site within this region can then be derived.

Multiple regression models and the index-flood method (IFM) are the prime methods for RFFA. The regression on quantile approach uses regression analysis to develop equations to relate climate and physiographic characteristics to the flow quantiles estimated from single-station flood frequency analysis in a homogeneous region. The index-flood method (IFM) establishes a relationship (growth curve) between the

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scaled quantiles and the return period in a homogeneous region. Regionalization, substituting space for time, is regarded as the fundamental premise of RFFA. The L-moments based index-flood method is an advanced approach which has been widely used for flood studies. Recent studies include the regional flood frequency analysis in Sicily, Italy by Noto (2009); the regional flood estimation for ungauged basins in Sarawak, Malaysia by Lim & Lye (2003); and the regional flood frequency analysis for West Mediterranean Region of Turkey by Saf (2009). L-moments are the linear combination of PWMs. Because its parameters are less biased, it has the ability to estimate site characteristics in a simple way; in particular, to estimate distribution parameters. Detailed information about L-moments is presented in Chapter 2.

In general, the application of the IFM should satisfy two assumptions: 1) the data at each site are independent and identically distributed; 2) the frequency distribution at each site should be identical except for the scale factor. Based on Hosking and Wallis (1997), the index-flood method based on L-moments has the following steps:

1) Screening the data: The objective is to check for gross errors of the data and to make sure the data is continuously available over time. That is, there is no gap or missing data.

2) Identifying the homogeneous region: Deciding on which river basins can be grouped together as a homogeneous region. That is, flood data with approximate identical distribution except for scale.

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3) Choosing a frequency distribution: Since the regional frequency distribution is essentially determined by the L-moment ratio diagram, a goodness-of-fit test will determine how well the selected distribution fit the data in the region. The application of robustness test can become necessary when there is more than one acceptable regional frequency distribution.

4) Estimating the frequency distribution: This process is designed to compute the flood quantiles for certain return periods at ungauged sites derived from the regional growth curve.

1.2 The application of RFFA for Newfoundland and Labrador

The first regional flood frequency analysis for Newfoundland was performed by Poulin (1971). Subsequent to this, regular updates by the provincial government of Newfoundland were carried out in 1984, 1990, 1999 and 2014 (Government of Newfoundland and Labrador, 1984; Government of Newfoundland and Labrador, 1990; Government of Newfoundland and Labrador, 1999 & AMEC, 2014). The regression method as described earlier, based on the observed data and sites characteristics, was the prime methodology used. However, this methodology, while easy to understand and apply often suffers from lack of consistent results and accuracy due to short historical data and other statistical issues.

The first RFFA for Newfoundland based on the L-moments index-flood procedure

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was proposed by Pokhrel (2002). The regional divisions in this analysis were based on two references: 1) the division of four sub regions (A, B, C and D) used in the provincial government analysis in 1989, and 2) the division of two sub regions Y and Z suggested by the Water Survey of Canada (WSC). This research concluded that the WSC sub regions obtained more accurate quantile estimations than sub regions suggested in 1989. The generalized extreme value (GEV) distribution was also found to have a superior performance compared to the lognormal (LN3) distribution for the regions of 1989. The comparison between the at-site and regional estimates showed that the L-moments based IFM has the ability to provide more accurate quantile estimation for the ungauged sites than conventional regression models, and it obtained more accurate results than the study in 1989.

1.3 Rationale and objectives

As introduced in Section 1.2, the L-moments based RFFA had been successfully applied in in Newfoundland and Labrador, therefore, in this thesis, due to the excellent performance and the worldwide application of the L-moments based index-flood approach, the regional flood frequency analysis for the Island of Newfoundland will be updated with the latest data up to 2013. For Labrador, a RFFA using the L-moments based index-flood approach will be used for the first time to obtain flood quantile estimates for ungauged basins. These results will be compared

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to those based on the regression based approach recently completed by AMEC (2014). The two main objectives of this thesis can thus be summarized as follows:

1) Update the quantile estimates at both gauged and ungauged sites for the Island of Newfoundland via the L-moments based index-flood procedure of RFFA. The updated results will be compared to those obtained by Pokhrel (2002) and those recently obtained by AMEC (2014).

2) Develop the first regional flood frequency analysis for Labrador using the L-moments based index-flood method and compare the results with those obtained using the regression method developed by AMEC (2014).

1.4 Outline

This thesis has six chapters. Chapter 1 provides a general introduction to regional flood frequency analysis in Newfoundland and Labrador and the methodologies used.

It also provides objectives for the study and an outline of the thesis. Chapter 2 reviews recent and related research in the field of regional flood frequency analysis and application of RFFA in Newfoundland and Labrador. Popular methodologies used for RFFA, the application of index-flood procedure in hydrologic research, the commonly used methods of fitting frequency distribution models and methods for identifying homogeneous regions are also discussed. Chapter 3 describes the

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methods used for the L-moments based index-flood procedure in a step by step manner. Chapter 4 presents the results of RFFA in Labrador and estimates the index flood at ungauged sites using a nonlinear regression model. A comparison with results from AMEC (2014) will also be presented. Updated results of the RFFA for the Island of Newfoundland will be shown in the Chapter 5, as well as the comparison of quantile results with those of Pokhrel (2002). Chapter 6 summarizes the results and provides conclusions and recommendations regarding the application of the L-moments based index-flood of RFFA in Newfoundland and Labrador.

Limitations of this research are also discussed. A list of the references and programming codes used in this thesis are presented as appendices.

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CHAPTER 2

LITERATURE REVIEW

2.1 General

This chapter first reviews some of the key steps in regional flood frequency analysis and some of the literature for each step. This is then followed by a brief review of RFFA that have been conducted in Newfoundland and Labrador. The estimation of regional flood frequency equations or curves has always been a popular topic among hydrologists and even among some statisticians. Two main methodologies in use today for RFFA are: 1) regional quantile regression approach which became popular with the advent of computers for performing multiple regression analysis; and 2) the index-flood approach which describes a regional quantile growth curve estimated graphically or by statistical methods. Since the development of the statistically sophisticated L-moments index-flood approach by Hosking and Wallis (1997), this approach is perhaps the most widely used worldwide today. Since the objectives of this thesis involve the use of the L-moments based index-flood procedure, the steps involved with this approach will be followed and literature pertaining to each step reviewed. RFFA using the L-moments based index-flood approach is carried out based on the following six steps:

1) Screen the data;

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2) Define a homogeneous region;

3) Perform the homogeneity test for each proposed region;

4) Select a regional frequency distribution for each region and check for robustness;

5) Estimate flow quantiles for both gauged and ungauged sites; and

6) Verify and assess accuracy of quantile estimation.

In the next section, the literature review is based on the steps mentioned above.

2.2 Screening the data

Data screening is the first step to be taken in any data analysis. Before starting the work, one should ensure that the data is appropriate for the analysis. Questions to be asked include: 1) Are the environmental data of sufficient quality and quantity and do they follow the same frequency distribution? 2) Have the data changed over time?

For example, the data used in this thesis may have been adjusted or corrected by the Water Survey of Canada (WSC). It is possible that the WSC did not just update the historical data to the year 2013; they may have also modified the record for every year at each station. For RFFA, Hosking and Wallis (1997) suggested three kinds of useful checks: 1) check the data individually to find gross errors (Wallis, et al. 1991);

2) check the data at each site for outliers and repeated values; 3) check for trends and

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abrupt changes and compare the data between sites.

The existence of extreme values or outliers may bring bias to the estimation, but to simply discard the outliers may distort results (Kirby, 1974). Therefore, tests become necessary to screen out outliers and then to check whether they can be accepted within a homogeneous group. There are many tests for outliers. For example, the U.S.

Water Resource Council (1981) used a statistical hypothesis test in flood frequency estimation, which compared the difference between the outliers and other values in a sample. A method based on a so-called “masking effect” was applied successfully by Barnett and Lewis (1994), which had an ability to distinguish multiple outliers. The sum of square statistics (Grubbs, 1950) and extreme-location statistics (e.g. Epstein, 1960a & 1960b) are other tests for outliers. Hosking and Wallis (1997) reported that double-mass plots or quantile-quantile plots are also well-known methods for detecting outliers which are easy to apply. Boxplot, histogram plot and dot plot provided in some statistical software can also work well for detecting outliers.

Another alternative method is the L-moment ratios (Hosking and Wallis, 1997), which is designed to detect unusual sites from a group of sites by comparing their individual L-moment ratios with the regional L-moment ratios of a group. The detailed principle and application of L-moments will be discussed later in Chapter 3.

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2.3 Definition of a homogeneous region

For RFFA it is common that the data of some sites are insufficient to provide reliable estimation. Therefore, identifying a homogenous region is a good way to transfer information from other available neighboring stations. Hosking and Wallis (1997) summarized some commonly used grouping methods to decide on a homogeneous region.

2.3.1 Geographical convenience

Delineating a homogeneous region based on geographical convenience is a direct and traditional method. The definition of geographical convenience usually means the administrative area (Natural Environment Research Council, 1975; Beable and McKerchar, 1982), political or physiographic boundaries. However, for larger areas the variability of physical or physiographic site characteristics may be large;

therefore, the identification of a homogeneous region simply depending on geographical parameters is rarely used in recent studies. Attempts to define a homogeneous region based on geographical parameters are usually accompanied with a goodness-of-fit test or hypothesis test to make sure the defined sub-regions are reasonable and unbiased for RFFA.

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2.3.2 Clustering techniques

Cluster analysis can be hierarchical and non-hierarchical (Downs & Barnard, 1992).

It is a very developed and widely used technique of dividing a data set into groups or to combine several data sets into a group based on similar data vectors (site characteristics or at-site statistics) (Hosking and Wallis, 1997). This technique has been used in many hydrological studies worldwide. For example, Jingyi and Hall (2004) used Ward Linkage clustering, in addition to Fuzzy C-Means and Kohonen neural network to successfully delineate homogenous regions in the southeast of China. Hierarchical clustering analysis was used for regional estimation in Mexico (Quarda, T et al. 2008); and Bharath (2015) completed the delineation of homogeneous regions in India using wavelet-based global fuzzy cluster analysis.

Burn (1989) identified homogeneous regions by combining cluster analysis and basin similarity measures. The hybrid-cluster and K-means algorithm was recommended by Rao & Srinivas (2006) for regionalization in Indiana, USA; and a fuzzy clustering approach was applied by Srinivas et al. (2008) to identify the regions of watersheds for flood frequency analysis. Noto and Loggia (2009) divided five regions using cluster analysis in Sicily, Italy; and Luis-Perez et al. (2011) applied two kinds of clustering techniques to delineate homogeneous regions in the Mexican-Mixteca region while Basu, B. and Srinivas, V. V. (2014) used kernel-based fuzzy clustering analysis to identify the homogeneous groups of watersheds in the U.S. Other related studies that used cluster analysis include Shu and Burn (2004), Wiltshire (1986),

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Bhaskar and O’Connor (1989), Mosley (1981), Tasker (1982), Nathan and McMahon (1990), Richman and Lamb (1985), Kalkstein et al. (1987), Burn and Goel (2000), Lim and Lye (2003), and Fovell & Fovell (1993).

2.3.3 Subjective partitioning

This method of delineating homogeneous regions requires the sites to have similar site characteristics. Similar site variables include the amount of rainfall, drainage area, timing of floods, forested areas, etc. Gingras, Adamowski and Pilon (1994) used the time of year when the largest flood occurred as the parameter to delineate sub regions in Ontario and Quebec. De Coursey (1972) formed groups of basins with similar flood responses in Oklahoma. For RFFA, a heterogeneity test is usually carried out after using a subjective partitioning method. But as Hosking and Wallis (1997) mentioned, when the at-site statistics are used as the basis for subjective partitioning, the validity of the use of the heterogeneity measure may be affected in validating the regions.

2.3.4 Objective partitioning

This method is designed to divide sites into groups depending on whether their site characteristics exceed one or more threshold values. Mailhot et al. (2013) applied the

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approach of Peak-Over-Threshold (POT) to estimate intense rainfall in southern Quebec. Pearson (1991b) used this procedure and successfully analyzed small basins’

grouping in New Zealand. Similar to subjective partitioning, it is recommended that heterogeneity tests be carried out for the delineated regions when using this method of partitioning (Hosking and Wallis, 1997).

2.3.5 Other grouping methods

Other alternative methods of defining homogeneous regions include the method of residuals, canonical correlation analysis, and region-of-influence (ROI) (Basu, 2014), among others. For example, White (1975) grouped basins based on the factor analysis of the site characteristics in Pennsylvania; and Burn (1988) applied principal components analysis to group gauged sites, depending on which subjectively rotated set of principal components a site’s annual maximum streamflow most closely resembled.

2.4 Homogeneity test for proposed regions

A homogeneous region is a fundamental requirement for quantile estimation. Once the regions or sub-regions are identified, a homogeneity test is needed to make sure that the delineated regions and subsequent analysis are appropriate and meaningful.

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Multiple methods have been used for testing the degree of homogeneity of a region.

Dalrymple (1960) proposed the first test that fitted the Gumbel distribution as the underlying distribution to every studied site. Chow (1964) tested the homogeneity by analyzing the sample coefficients of variation (Cv) and /or skewness (Cs). Lu (1991) used the L-moment ratios and normalized 10-year flood estimate to conduct a regional homogeneity test. Lu and Stedinger (1992) carried out a homogeneity test based on the sample variance and normalized 10-year flood quantile estimators.

Fill and Stedinger (1995) compared the power of the Dalrymple test, normalized quantile test and a method of moment Cv test. Scholz and Stephen (1987) proposed the Anderson-Darling test (Anderson and Darling, 1954) for testing the homogeneity of samples. However, the most popular method is the L-moment ratios based heterogeneity test proposed by Hosking and Wallis (1993, 1997), which has been widely used in hydrological studies. For example, Gabriele and Chiaravalloti (2012) used this method to test the degree of homogeneity based on the rainfall sample data within regions. Abolverdi and Khalili (2010) tested the degree of homogeneity in southwestern Iran based on the regional rainfall annual maxima, among many other studies. The detailed formulation of this test will be discussed in Chapter 3.

2.5 Selection of regional frequency distribution

The appropriate selection of a regional flood distribution has a direct impact on the

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quantile estimation at gauged and ungauged sites. A regional frequency distribution is fitted from a single site to other sites within the homogeneous region. As Hosking and Wallis (1997) mentioned that there may be more than one acceptable candidate regional distributions, and the best fitting distribution is the one with the ability to reflect the “true” distribution. Therefore, rather than identifying a “true” distribution, the aim is to determine a distribution which will provide the most approximate fit to the observed data and yield a more accurate quantile estimation for each single site.

Sveinsson (2001) compared the quantile estimation based on the population index flood fitted by the GEV distribution using Hosking and Wallis’s (1997) index flood regional PWM procedure. The Log Pearson Type (Ⅲ) was used for peak flood discharge based on Bulletin 17 B in the U.S. (Lim and Voeller, 2009). Ashkar and Quarda (1996) discussed the use of the generalized Pareto distribution for flood frequency analysis. Griffis and Stedinger (2007) fitted LN3 distribution to the flood quantile estimation using the weighted Bulletin 17B procedure. Peel et al. (2001) compared multiple distributions based on two graphical different methods and found that using graphical methods with an L-moment ratio diagram can distort the choice of regional distribution of observed data. In a regional flood frequency analysis of the west Mediterranean region of Turkey, Saf (2009) found that Pearson type Ⅲ distribution fitted well to the Antalya and lower-west Mediterranean, and that Generalized Logistic distribution was most suitable for the upper-west Mediterranean.

In a Canada-wide study, Yue and Wang (2004a, b) fitted the generalized extreme

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value (GEV) for the Pacific and southern British Columbia mountains, the 3-parameter lognormal distribution to the northwestern forest area, the Wakeby distribution to the Arctic tundra, the Pearson type Ⅲ to the Prairies, Northeastern forest, Great Lakes, and regions in St. Lawrence, Atlantic and Mackenzie. Atiem and Harmancioglu (2006) fitted five different distributions---generalized Pareto (GPAR), generalized extreme value, generalized logistic, generalized normal and PE3 to the annual maximum flood data for 14 sites in the Nile River tributaries based on the index-flood method.

The application of the moment-ratio diagram introduced by McCuen (1985) provides a quick and basic approach to judge how candidate distributions fit the data. Hosking (1990) recommended using L-moments which is a linear combination of the ranked observed data and exhibits less bias than the traditional moments. The L-moment ratio diagram is a simple plot of τ4 against τ3 (L-kurtosis and L-skewness) for commonly used distributions, and the at-site and regional average L-moment ratios can be plotted to compare with the population values of commonly used distributions (Hosking and Wallis, 1997). Since the use of the L-moment ratio diagram is a quick and basic approach to select a regional distribution, the final determination must rely on further goodness-of-fit and robustness tests. Goodness-of-fit tests include quantile-quantile plots, Kolmogorov-Smirnov, chi-squared and the most popular L-moments based tests introduced by Hosking and Wallis (1997).

The L-moment ratios based goodness-of-fit test is designed to test whether a given

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regional distribution can provide a close fit to the data using a simulation process.

Using parts of the approaches mentioned above, Malekinezhad (2011) determined that the generalized extreme value distribution was the best fit for flood estimation in the Namak-Lake basin. Atiem and Harmancioglu, (2006) found that the generalized logistic distribution provided the best fit for the data in the River Nile. Mkhandi and Kachroo (1997) found that the Pearson type Ⅲ was the most suitable distribution for regional flood in southern Africa. In another study, the Generalized Normal distribution was identified as the best fit for the flood data in the Mahi-Sabarmati Basin (Parida, et al. 1998). If there is more than one acceptable distribution, the robustness test (to be described in Chapter 3) is suggested when the underlying distribution is different from the selected one.

2.6 Quantile flow estimation for both gauged and ungauged sties

The index-flood procedure plays a key role in the estimation of flow quantiles. Once the studied region is found to be homogeneous and the regional frequency distribution has been determined, and it is assumed that the frequency distribution of all sites in the region is identical except for the site-special scaling factor known as the index flood. The index flood is usually the mean annual flood or the median annual flood. The flow quantiles can be estimated as the product of the index flood and regional growth curve or regional frequency distribution function. Early

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applications of the index-flood procedure include Dalrymple (1960) and NERC (1975), Hosking et al. (1985), Jin and Stedinger (1989), Wallis and Wood (1985), Letttenmaier and Potter (1985). Cunnane (1988) and Pitlick (1994) demonstrated successful applications of the index-flood procedure for regional flood frequency analysis. Madsen et al. (1997) illustrated the advantages of the use of index-flood procedure in terms of both annual flood series and partial duration series. Portela and Dias (2005) described six homogeneous regions and used the data of annual maximum flood series of 120 Portuguese stream gauging stations in mainland Portugal using the index-flood method. Later, Hosking and Wallis (1997) successfully introduced the use of L-moments in the index-flood procedure and it was shown to be robust in the presence of any extreme values and outliers. Recent regional flood studies based on index-flood procedure include studies in the U.S.A.

(Vogel et al. 1993; Vogel and Wilson, 1996), Malaysia (Lim and Lye, 2003), Australia (Pearson et al. 1991), Southern Africa (Mkhandi and Kachroo, 2000), New Zealand (Pearson, 1991, 1995; Madsen et al, 1997) and Turkey (Saf et al. 2009).

Estimation of flow quantiles at gauged sites with short records can be completed directly from an estimate of the index flood using the annual maximum or peaks-over-threshold values. For the ungauged sites where their index floods are not available, the most commonly used method is a regression model. The index flood of gauged sites is regressed against their respective catchment or site characteristics (e.g., basin area, length, basin slope, drainage density, etc.) to obtain a model relating

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the index flood to basic characteristics. For example, IH (1999) developed regression models relating median flow to five different site characteristics. Brath et al (2001) reviewed different indirect methods of estimating the index flood at gauged sites and concluded that the regression method had a better performance than other approaches.

The regression method was used for regional flood frequency analysis in Sicily (Noto and Loggia, 2009) and regional flood estimation for ungauged basins in Sarawak, Malaysia (Lim and Lye, 2003), among many others.

2.7 Verification and assessment of accuracy of quantile estimation

Accuracy of assessment is always needed for model evaluation. The factors that have an influence on accuracy of assessment are: 1) the regions are not adequately homogeneous, 2) the regional frequency distribution is not robust, and 3) the availability of data is limited. Assessment accuracy based on traditional statistics involves constructing confidence intervals for estimated parameters or quantiles on the assumption that all the statistical assumptions of the models are satisfactory.

However, in practice, it is found that in most cases it is difficult to ensure that the models used are the “correct” ones (Hosking and Wallis, 1997). Although it is difficult to establish common criteria for model evaluation, the evaluation is still carried out in some studies based on some specific statistics such as sensitivity analysis and model calibration. For example, Gupta et al. (1999) calibrated

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hydrological models using the shuffled complex evolution automatic procedure;

Motovilov, et al. (1999) verified hydrological model ECOMAG with the use of standard meteorological and hydrological data in the NOPEX southern region; Van Liew et al. (2007) used two sub watersheds in the Little Washita River Experimental Watershed (LWREW) to calibrate the parameters of the Soil and Water Assessment Tool (SWAT) and Hydrologic Simulation Program-Fortran (HSPF) models in southwestern Oklahoma. Other similar research includes Santhi et al. (2001) and Singh et al. (2004). However, nobody used the acceptable ranges of values for each statistic until a review of the values for various statistics used was provided by Borah and Bera (2004).

A good model evaluation entails satisfying the following conditions: 1) it must be robust and acceptable to various constituents and climatic conditions, 2) be commonly used and recommended by various studies, and 3) be robust in model evaluation (Moriasi et al. 2007). Boyle et al. (2000) recommended the estimation of residual variance (the difference between the measure and simulated values) which can be estimated by the residual mean square or root mean square error (RMSE).

Hosking and Wallis (1997) suggested the Monte Carlo simulation approach to assess the accuracy by calculating the RMSE when the region is not homogeneous enough, the regional frequency distribution is misspecified or the observed data are statistically dependent. Chapter 3 provides detailed information of the Monte Carlo simulation approach based on L-moments. The assessment of RMSE has been

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widely applied in various studies; for example, Fill and Stedinger (1998) showed that the empirical Bayes estimator had the same or a better performance than the simpler normalized quantile regression estimator for sites with shorter records based on the results of RMSE; Saf (2009) developed a Monte Carlo simulation and evaluated the accuracy of the quantile estimates based on the relative root-mean-square error and relative bias; and Atiem and Harmancioglu (2006) evaluated the results of quantile estimation by assessing the RMSE% which is also based on the Monte Carlo simulation approach.

Other methods for example, the slope and y-intercept of the best-fit regression line, can indicate how the simulated data match the observed data on the assumption that the observed and simulated data are linearly related (Moriasi et al. 2007). The use of Pearson’s correlation coefficient (r) and coefficient of determination (R²) to measure the degree of linear collinearity between simulated and observed data are also popular; The index of agreement (d) (Willmott, 1984) that measures the degree of model prediction error and varies between 0 and 1 and Nash-Sutcliffe efficiency (NSE) that is designed to compare the relative magnitude of the residual variance (“noise”) to the measured data variance (“information”) (Nash and Sutcliffe, 1970) are also widely used ; Persistence model efficiency (PME)-a normalized model that evaluates the relative magnitude of the residual variance (“noise”) to the variance of the errors obtained using a simple persistence model (Gupta et al, 1999) and the Prediction efficiency (Pe) (Santhi et al. 2001) that can determine how well the

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simulated data can fit the observed data are also possible.

2.8 RFFA for Newfoundland

RFFA for the Island of Newfoundland was conducted in 1971, 1984, 1989, 1999, 2002 and 2014 by the Government of Canada, Newfoundland or its consultants (Poulin, 1971; Government of Newfoundland, 1984; Government of Newfoundland, 1990, Government of Newfoundland and Labrador, 1999; Pokhrel, 2002 & AMEC, 2014). The study in 2014 was the first to include the Labrador. The first provincial flood frequency research was by Poulin (Government of Canada, 1971) used the classical index-flood approach. In that study, the Island of Newfoundland was analyzed as one region with 17 gauged stations. The index flood which was the mean flows was used to develop a function relating the mean (Q) and the drainage area (DA).

In a subsequent study (Government of Newfoundland, 1984), the Island of Newfoundland was sub-divided into two regions, a North and a South region. Twenty one gauged stations were analyzed based on the regression on quantiles approach.

Single-site flood frequency was performed for each station to obtain estimates of several key quantiles. Then these were regressed against site characteristics such as drainage area (DA) and latitude for the North region, and drainage area (DA), area controlled by lakes and swamps (ACLS) and slope in the South region. However,

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Lye and Moore (1991) noted that the log-transformed model and the use of mean flow as the predictor value were not properly carried out which would lead to bias, and the variable latitude was not suitable for North region given that it is a very small area.

A new regional flood frequency analysis conducted by the provincial government in 1989 increased the number of gauged stations up to thirty-nine. This study divided the Island of Newfoundland into four regions (A-Avalon and Burin Peninsula;

B-central region of the Island; C-Humber valley and northern peninsula; and D-the southwestern region of the island) taking into account the availability of data, the timing of regional floods and physiographic factors such as flood characteristics, amount of precipitation and results of regression analysis. The average record length was 21 years and the record was extended in some stations with short records. The drainage area (DA), lakes and swamps factor (LSF), drainage density and slope were included for the regression on quantiles.

In an updated study by the Government in 1999, the four sub regions of 1989 were renamed. The new names are -northwest (NW), northeast (NE), southeast (SE) and southwest (SW) and refer to the previous C, B, A, and D regions proposed in 1989.

The drainage area (DA) and lake attenuation factor (LAF) were found to be significant predictors and the LSF was a significant variable only in the SW region.

Instead of the regression on quantiles approach, Pokhrel (2002) conducted a regional

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flood frequency analysis of the Island of Newfoundland using annual peak flow until 1998 based on the L-moments index-flood procedure suggested by Hosking (1990).

The Island of Newfoundland was sub-divided based on two kinds of regionalization -four regions as used by the Government of Newfoundland in 1989 and 1999, and the Y and Z regions suggested by the Water Survey of Canada (WSC). The determination of homogenous regions, selection of regional frequency distribution and quantile estimation were all based on the L-moments based index-flood method.

It was found that for sub region Y, only the drainage area (DA) and drainage density (DRD) were significant at α=5% in terms of estimating the index flood. For sub region Z, in addition to DA and DRD, the lakes and swamps factor (LSF) was significant as well. The study also showed that the L-moments based index-flood approach with the Y and Z regions was superior to that of the regression on quantile approach and use of four sub-regions.

The latest provincial RFFA (AMEC, 2014) was conducted for both Newfoundland and Labrador. This study also used the regression on quantile approach, but with data up to 2012. Hence newly updated regression models were obtained. Seventy-eight gauged stations in Newfoundland and twelve gauged stations in Labrador were used in the study. Regression equations were obtained considering the Island of Newfoundland as a single homogeneous region and considering it as four sub hydrological homogeneous regions as proposed in 1999. Drainage area (DA) and lake attenuation factor (LAF) were significant for the NW, SE and NE sub regions,

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whereas the lakes and swamps factor (LSF) and DA were shown to be significant for the SW region. Labrador was analyzed as a single homogeneous region and only DA was significant for developing regression equations. The study did not compare results to those obtained by Pokhrel (2002) nor were any robustness tests conducted.

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CHAPTER 3

METHODOLOGY

3.1 General

The index-flood method (IFM), a widely used regression method for regional frequency flood analysis, was first proposed by the U.S. Geological Survey (Dalrymple, 1960). Many successful applications show that the IFM has the ability to define a more reliable homogeneous region in which the variability of the at-site data at gauged sites objectively exists. The quantile estimation at each gauged site can be derived directly from the regional flood quantile function, even for the ungauged sites. Hence determining the flood quantiles within a defined homogeneous region is possible.

The detailed modern procedures of the IFM suggested by Hosking (1990) and Hosking and Wallis (1997) will be introduced in this chapter. The L-moments, a modern and advanced mathematical statistics approach are involved to determine the homogenous regions, selection of the regional flood frequency distribution and quantile flows for both gauged sites and ungauged sites.

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3.2 Regional flood frequency analysis

The development of a regional flood frequency analysis (RFFA) has proved to be an effective method for estimating flood quantiles at ungauged sites or sites with insufficient streamflow data using the flood information at neighboring sites within a homogeneous region. Compared to the traditional at-site estimations, the regional data can minimize the standard error of interest. Regional regression models and the index-flood procedure are commonly used for RFFA in previous and recent flooding studies. The regression approach develops regression equations to relate at-site climate and physiographic characteristics to flow quantiles from each single site within a homogeneous region. However, the regional regression approach sometimes has a limited ability to provide reliable estimations when the numbers of gauged stations are insufficient. Uncertainties and bias are inevitable. The modern index-flood procedure however successfully avoids these disadvantages. Instead, the flood quantiles at gauged sites can be achieved based on relationship between the quantile function of the regional frequency distribution and index flood at each site.

Even for the ungauged sites, the quantiles estimation can be easily achieved using estimated index flood. According to Hosking and Wallis (1997) the quantile estimates can be obtained from:

Qi(F)=u_i*q(F) i=1, 2, 3…N [3.1]

where u_iis the index flood at sites i in a homogenous region with N sites and q(F) is

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the regional growth curve. The index flood at ungauged sites can be obtained by establishing a linear or nonlinear regression relationship between sites characteristics and the index flood at gauged sites within a homogeneous region. The application of the index-flood procedure follows an important assumption that all the sites in a defined region are distributed ideally except for a scale factor. Multiple recent studies show that index-flood approach can produce a more accurate and reliable quantile estimation than the regression on quantile approach (e.g. Pokhrel, 2002; Noto, 2009;

Lim & Lye, 2003; Saf, 2009).

3.3 L-moments

L-moments are the linear combination of probability weighted moments (PWMs) which is widely used in fitting frequency distribution, estimating distribution parameters and hypothesis testing in flood frequency analysis. Greenwood et al.

(1979) defined the PWMs as:

βr=E{X [F(x)^r]} [3.2]

where F(x) is the cumulative distribution function for X. X(F) is the inverse CDF of X evaluated at the probability F. βr equals to the mean stream flow when r=0.

Later, Hosking (1990) modified the “probability weighted moments” as:

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βr=¹_n ∑^n−r_j=1[₍⁽n−1^n−j^r ⁾

r )] x(j) r=0, 1, … , [3.3]

where x(j) is the ordered stream flow; r is the probability weighted moments; n is the sample size, and j is the order of the observed steam flow.

The L-moments are generally defined as Eq. [3.4]. The first four L-moments are the mean of distribution, measure of scale, measures of skewness and kurtosis respectively, which are defined in Eq. [3.5].

λr+1=∑^r_k=0(−1)^r-k(_k^r)(^r+k_k )βr r=0, 1, … , [3.4]

λ1=β0 λ2=2β1-β0 λ3=6β2-6β1+β0 λ4=20β3-30β2+12β1-β0 [3.5]

Additionally, the dimensionless L-moments called L-moments ratios including L-CV, L-skewness and L-kurtosis shown in Eq. [3.6] also play key roles in the estimations of parameters of candidate distributions and the determination of the regional flood frequency distribution. In particular, the L-moment ratio diagram, plot of sample L-moment ratios, average L-moment ratios and theoretical L-moment ratios curves of candidate distributions on a single graph provides an essential visual tool to distinguish among the candidate distributions.

L-CV=λ2/λ1 L-skewness(τ3)=λ3/λ2 L-kurtosis(τ4)=λ4/λ2 [3.6]

The applications of L-moments show great advantages over conventional moments

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(C-Moments). Hosking (1990) concluded that L-moments and L-moment ratios are nearly unbiased even for highly skewed observations. They have less sensitivity to the sample size and extreme observations and are more robust to outliers. The L-moment ratio diagram had been shown to be a useful tool to distinguish among candidate distributions by plotting sample L-moment ratios (L-skewness and L-kurtosis) and comparing them with theoretical L-moment ratios curves of candidate distributions. The theoretical L-moment ratios curves of commonly used candidate distributions on the L-moment ratio diagram are shown in Figure 3.1.

GPA- generalized Pareto; GEV- generalized extreme-value; GLO- generalized logistic; LN3- lognormal; OLB- overall lower bound of τ₄ as a function of τ₃; and PE3- Pearson typeⅢ.

Figure 3.1 L-moment ratio diagram (after Hosking and Wallis, 1997)

-0.1 0 0.1 0.2 0.3 0.4 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

GPA GEV GLO LN3 PE3 OLB

L-kurtosis

L-skewness

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Compared to PWMs, the applications of L-moments are more convenient and simpler for measuring the shape and scale of the observations. As noted in the introductory chapter, the applications of L-moments for the index-flood procedure are represented by the following steps (Hosking and Wallis, 1997):

1) Screening the data and use of discordancy measure;

2) Plotting sample L-moment ratios on Figure 3.1 to select a tentative regional frequency distribution;

3) Utilizing the regional homogeneity test based on Monte Carlo simulation to test the homogeneity of the region, and

4) Applying the goodness-of-fit test and robustness tests to determine the final regional flood frequency distribution.

3.4 Procedures for the index-flood based RFFA

Hosking and Wallis (1997) suggested that the application of index-flood based RFFA should follow the key assumption that all of the observations within a defined homogenous region are ideally distributed except for a scale factor (index flood). The procedure of index flood estimation uses the following steps:

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1) Screening the data and discordancy measure.

2) Definition of a homogeneous region.

3) Selection of a regional frequency distribution, and

4) Quantile flow estimation for both gauged and ungauged sites.

3.4.1 Screening the data and discordancy measure

The data used for the RFFA is required to represent the true quantity being measured, and all of the observations should follow the same distribution. Basically, the purpose of screening the data is to satisfy three requirements: 1) the data collected for analysis are correct, 2) there are no extreme values or outliers, and 3) the data did not change over time. Hosking and Wallis (1993) first proposed the L-moments based discordancy measure (Eq. [3.7]) to identify unusual sites with different L-moment ratios from other sites within a region. The discordancy measure can be calculated using the Matlab program code (Appendix A-1).

D_i= ¹

3N(u_i− u̅)^T A⁻¹(u_i− u̅) [3.7]

where Di is the discordancy measure, ui =[t⁽ⁱ⁾ t3(i)

t4(i)

]^Tis a vector of t, t3 and t4 for site i in a region with N sites and u̅ is the unweighted group average which can be defined as:

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u̅ =N⁻¹ ∑^N_i=1u_i [3.8]

And A is the covariance matrix ofu_i, given by

A=∑ (u^N_i=1 _i− u̅)(u_i− u̅)^T [3.9]

Applying this measure, the unusual sites with inconsistent L-moments ratios due to incorrect records or gross error can be screened out; then for the unusual sites, they might be removed or be included in another region based on the further investigation.

Hosking and Wallis (1997) stated that the conclusion reached based on the discordancy measure largely depends on the number of sites in a region. Generally, the algebraic bound of Di should satisfy:

Di ≤ (N-1)/3 [3.10]

The sites can be regarded as discordant from the remaining sites if the Di value is larger than the critical value shown in Table 3.1. They also suggested that the Di ≥3 is only suitable for regions with 11 or more sites.

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Table 3.1 Critical values of discordancy measure with N sites (Hosking and Wallis, 1997)

Number of sites in a

region Critical value Number of sites in a

region Critical value

5 1.333 6 1.648

7 1.917 8 2.140

9 2.329 10 2.491

11 2.632 12 2.757

13 2.869 14 2.971

>15 3

3.4.2 Delineation of homogeneous regions

The delineation of a homogeneous region is a prime step for regional flood frequency analysis. To determine whether a proposed region is homogeneous or not, Hosking and Wallis (1993) suggested a heterogeneity test which aims to assess the degree of homogeneity by comparing the between-site variations in sample L-moment ratios for the sites in a group with what the expected value would be in a definitely homogeneous region. The between-site variation of L-moment ratios is measured by calculating the standard deviation (Eq. [3.12]) of sample L-CVs.

The principle of heterogeneity test (Hosking and Wallis, 1997) can be described as:

assume a region has N sites. Each site has the record length of ni. t⁽ⁱ⁾, t3(i)

and t4(i)

represent the sample L-moment ratios respectively, of which the weighted regional average L-moment ratios are defined as:

LIST OF FIGURES

ABSTRACT

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF SYMBOLS

LIST OF ACRNYMS

CHAPTER 1

CHAPTER 2

CHAPTER 3